353:
407:
986:
928:
1040:
704:
848:
651:
578:
1088:
780:
532:
875:
605:
265:
238:
211:
184:
133:
106:
1060:
744:
724:
496:
462:
434:
157:
77:
53:
468:. (Note that it is possible that neither player has a winning strategy.) Thus every Choquet space is Baire. On the other hand, there are Baire spaces (even
270:
358:
1181:
1154:
1123:
933:
1202:
880:
991:
1207:
656:
1093:
790:
807:
610:
537:
1066:
1177:
1150:
1140:
1119:
749:
501:
56:
28:, who was in 1969 the first to investigate such games. A closely related game is known as the
1171:
1144:
441:
21:
853:
583:
243:
216:
189:
162:
111:
797:
469:
82:
25:
1045:
729:
709:
481:
447:
419:
413:
142:
62:
38:
786:. Every strong Choquet space is a Choquet space, although the converse does not hold.
1196:
794:
348:{\displaystyle U_{0}\supseteq V_{0}\supseteq U_{1}\supseteq V_{1}\supseteq U_{2}...}
1113:
437:
472:
1063:
136:
267:, etc. The players continue this process, constructing a sequence
402:{\displaystyle \bigcap \limits _{i=0}^{\infty }U_{i}=\emptyset }
1115:
Lectures on
Analysis: Integration and topological vector spaces
981:{\displaystyle \operatorname {cl} (V_{i})\subseteq V_{i-1}}
475:
ones) that are not
Choquet spaces, so the converse fails.
1176:. Springer Science & Business Media. pp. 43–45.
804:
are strong
Choquet. (In the first case, Player II, given
464:
in which Player II has a winning strategy is called a
1069:
1048:
994:
936:
883:
856:
810:
752:
732:
712:
659:
613:
586:
540:
534:, is defined similarly, except that Player I chooses
504:
484:
450:
422:
361:
273:
246:
219:
192:
165:
145:
114:
85:
65:
41:
923:{\displaystyle \operatorname {diam} (V_{i})<1/i}
1146:The Descriptive Set Theory of Polish Group Actions
1082:
1062:.) Any subset of a strong Choquet space that is a
1054:
1035:{\displaystyle \left\{x_{i}\right\}\to x\in V_{i}}
1034:
980:
922:
869:
842:
774:
738:
718:
698:
645:
599:
572:
526:
490:
456:
428:
401:
347:
259:
232:
205:
178:
151:
127:
100:
71:
47:
746:in which Player II has a winning strategy for
409:then Player I wins, otherwise Player II wins.
8:
1149:. Cambridge University Press. p. 59.
108:, is defined as follows: Player I chooses
1092:is strong Choquet. Metrizable spaces are
1074:
1068:
1047:
1026:
1003:
993:
966:
950:
935:
912:
897:
882:
861:
855:
831:
818:
809:
757:
751:
731:
711:
690:
677:
664:
658:
634:
621:
612:
591:
585:
561:
548:
539:
509:
503:
483:
449:
421:
387:
377:
366:
360:
330:
317:
304:
291:
278:
272:
251:
245:
224:
218:
197:
191:
170:
164:
144:
119:
113:
84:
64:
40:
1096:if and only if they are strong Choquet.
1104:
7:
699:{\displaystyle x_{i}\in U_{i},V_{i}}
416:that a non-empty topological space
363:
396:
378:
14:
1173:Classical Descriptive Set Theory
444:. A nonempty topological space
440:if and only if Player I has no
1013:
956:
943:
903:
890:
837:
811:
769:
763:
640:
614:
567:
541:
521:
515:
95:
89:
1:
843:{\displaystyle (x_{i},U_{i})}
646:{\displaystyle (x_{1},U_{1})}
573:{\displaystyle (x_{0},U_{0})}
240:, a non-empty open subset of
186:, a non-empty open subset of
1170:Kechris, Alexander (2012).
1083:{\displaystyle G_{\delta }}
478:The strong Choquet game of
1224:
1112:Choquet, Gustave (1969).
580:, then Player II chooses
159:, then Player II chooses
775:{\displaystyle G^{s}(X)}
607:, then Player I chooses
527:{\displaystyle G^{s}(X)}
213:, then Player I chooses
1203:Descriptive set theory
1084:
1056:
1036:
982:
924:
871:
844:
791:complete metric spaces
776:
740:
726:. A topological space
720:
700:
647:
601:
574:
528:
492:
458:
430:
403:
382:
349:
261:
234:
207:
180:
153:
129:
102:
73:
59:. The Choquet game of
49:
1094:completely metrizable
1085:
1057:
1037:
983:
925:
872:
870:{\displaystyle V_{i}}
845:
777:
741:
721:
701:
648:
602:
600:{\displaystyle V_{0}}
575:
529:
493:
459:
431:
404:
362:
350:
262:
260:{\displaystyle V_{0}}
235:
233:{\displaystyle U_{1}}
208:
206:{\displaystyle U_{0}}
181:
179:{\displaystyle V_{0}}
154:
130:
128:{\displaystyle U_{0}}
103:
74:
50:
1067:
1046:
992:
988:. Then the sequence
934:
881:
854:
808:
784:strong Choquet space
750:
730:
710:
657:
611:
584:
538:
502:
482:
448:
420:
359:
271:
244:
217:
190:
163:
143:
112:
101:{\displaystyle G(X)}
83:
63:
39:
30:strong Choquet game
1118:. W. A. Benjamin.
1080:
1052:
1032:
978:
920:
867:
840:
772:
736:
716:
696:
643:
597:
570:
524:
488:
454:
426:
399:
345:
257:
230:
203:
176:
149:
125:
98:
69:
45:
1208:Topological games
1055:{\displaystyle i}
739:{\displaystyle X}
719:{\displaystyle i}
653:, etc, such that
491:{\displaystyle X}
457:{\displaystyle X}
429:{\displaystyle X}
412:It was proved by
152:{\displaystyle X}
72:{\displaystyle X}
57:topological space
48:{\displaystyle X}
1215:
1188:
1187:
1167:
1161:
1160:
1139:Becker, Howard;
1136:
1130:
1129:
1109:
1089:
1087:
1086:
1081:
1079:
1078:
1061:
1059:
1058:
1053:
1041:
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1038:
1033:
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1008:
1007:
987:
985:
984:
979:
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954:
929:
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921:
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901:
876:
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873:
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841:
836:
835:
823:
822:
781:
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773:
762:
761:
745:
743:
742:
737:
725:
723:
722:
717:
705:
703:
702:
697:
695:
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682:
681:
669:
668:
652:
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649:
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603:
598:
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579:
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576:
571:
566:
565:
553:
552:
533:
531:
530:
525:
514:
513:
497:
495:
494:
489:
463:
461:
460:
455:
442:winning strategy
435:
433:
432:
427:
408:
406:
405:
400:
392:
391:
381:
376:
354:
352:
351:
346:
335:
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321:
309:
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296:
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283:
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266:
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202:
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185:
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134:
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131:
126:
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123:
107:
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99:
78:
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75:
70:
54:
52:
51:
46:
22:topological game
1223:
1222:
1218:
1217:
1216:
1214:
1213:
1212:
1193:
1192:
1191:
1184:
1169:
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1157:
1138:
1137:
1133:
1126:
1111:
1110:
1106:
1102:
1070:
1065:
1064:
1044:
1043:
1022:
999:
995:
990:
989:
962:
946:
932:
931:
893:
879:
878:
857:
852:
851:
827:
814:
806:
805:
801:
753:
748:
747:
728:
727:
708:
707:
686:
673:
660:
655:
654:
630:
617:
609:
608:
587:
582:
581:
557:
544:
536:
535:
505:
500:
499:
480:
479:
446:
445:
418:
417:
383:
357:
356:
326:
313:
300:
287:
274:
269:
268:
247:
242:
241:
220:
215:
214:
193:
188:
187:
166:
161:
160:
141:
140:
115:
110:
109:
81:
80:
61:
60:
55:be a non-empty
37:
36:
26:Gustave Choquet
12:
11:
5:
1221:
1219:
1211:
1210:
1205:
1195:
1194:
1190:
1189:
1182:
1162:
1155:
1141:Kechris, A. S.
1131:
1124:
1103:
1101:
1098:
1077:
1073:
1051:
1029:
1025:
1021:
1018:
1015:
1011:
1006:
1002:
998:
975:
972:
969:
965:
961:
958:
953:
949:
945:
942:
939:
919:
915:
911:
908:
905:
900:
896:
892:
889:
886:
864:
860:
839:
834:
830:
826:
821:
817:
813:
799:
771:
768:
765:
760:
756:
735:
715:
693:
689:
685:
680:
676:
672:
667:
663:
642:
637:
633:
629:
624:
620:
616:
594:
590:
569:
564:
560:
556:
551:
547:
543:
523:
520:
517:
512:
508:
487:
453:
425:
414:John C. Oxtoby
398:
395:
390:
386:
380:
375:
372:
369:
365:
344:
341:
338:
333:
329:
325:
320:
316:
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303:
299:
294:
290:
286:
281:
277:
254:
250:
227:
223:
200:
196:
173:
169:
148:
135:, a non-empty
122:
118:
97:
94:
91:
88:
68:
44:
13:
10:
9:
6:
4:
3:
2:
1220:
1209:
1206:
1204:
1201:
1200:
1198:
1185:
1183:9781461241904
1179:
1175:
1174:
1166:
1163:
1158:
1156:9780521576055
1152:
1148:
1147:
1142:
1135:
1132:
1127:
1125:9780805369601
1121:
1117:
1116:
1108:
1105:
1099:
1097:
1095:
1091:
1075:
1071:
1049:
1027:
1023:
1019:
1016:
1009:
1004:
1000:
996:
973:
970:
967:
963:
959:
951:
947:
940:
937:
917:
913:
909:
906:
898:
894:
887:
884:
862:
858:
832:
828:
824:
819:
815:
803:
796:
792:
789:All nonempty
787:
785:
766:
758:
754:
733:
713:
691:
687:
683:
678:
674:
670:
665:
661:
635:
631:
627:
622:
618:
592:
588:
562:
558:
554:
549:
545:
518:
510:
506:
485:
476:
474:
471:
467:
466:Choquet space
451:
443:
439:
423:
415:
410:
393:
388:
384:
373:
370:
367:
342:
339:
336:
331:
327:
323:
318:
314:
310:
305:
301:
297:
292:
288:
284:
279:
275:
252:
248:
225:
221:
198:
194:
171:
167:
146:
138:
120:
116:
92:
86:
66:
58:
42:
33:
31:
27:
23:
19:
1172:
1165:
1145:
1134:
1114:
1107:
788:
783:
782:is called a
477:
465:
411:
34:
29:
24:named after
18:Choquet game
17:
15:
438:Baire space
137:open subset
1197:Categories
1100:References
877:such that
850:, chooses
473:metrizable
1076:δ
1020:∈
1014:→
971:−
960:⊆
941:
888:
671:∈
470:separable
397:∅
379:∞
364:⋂
324:⊇
311:⊇
298:⊇
285:⊇
1143:(1996).
1042:for all
706:for all
795:compact
1180:
1153:
1122:
802:spaces
436:is a
355:. If
20:is a
1178:ISBN
1151:ISBN
1120:ISBN
930:and
907:<
885:diam
793:and
35:Let
16:The
1090:set
139:of
1199::
938:cl
498:,
79:,
32:.
1186:.
1159:.
1128:.
1072:G
1050:i
1028:i
1024:V
1017:x
1010:}
1005:i
1001:x
997:{
974:1
968:i
964:V
957:)
952:i
948:V
944:(
918:i
914:/
910:1
904:)
899:i
895:V
891:(
863:i
859:V
838:)
833:i
829:U
825:,
820:i
816:x
812:(
800:2
798:T
770:)
767:X
764:(
759:s
755:G
734:X
714:i
692:i
688:V
684:,
679:i
675:U
666:i
662:x
641:)
636:1
632:U
628:,
623:1
619:x
615:(
593:0
589:V
568:)
563:0
559:U
555:,
550:0
546:x
542:(
522:)
519:X
516:(
511:s
507:G
486:X
452:X
424:X
394:=
389:i
385:U
374:0
371:=
368:i
343:.
340:.
337:.
332:2
328:U
319:1
315:V
306:1
302:U
293:0
289:V
280:0
276:U
253:0
249:V
226:1
222:U
199:0
195:U
172:0
168:V
147:X
121:0
117:U
96:)
93:X
90:(
87:G
67:X
43:X
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